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The document presents a derivation of modified shallow water equations (mSV) that account for significant variations in seabed topography from a variational principle. The mSV equations extend the classical Saint-Venant shallow water equations to be non-dispersive and non-hydrostatic while maintaining hyperbolic structure. Numerical methods are developed for the mSV system, including a finite volume scheme. Test cases demonstrate the ability of mSV to better model wave propagation over oscillating bottoms and tsunami waves compared to the standard shallow water equations.
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